A noncommutative Amir-Cambern theorem for von Neumann algebras and nuclear ${C}^∗$-algebras.

arXiv:1108.1970v2 [math.OA] 24 Jun 2013

A noncommutative Amir-Cambern theorem for von Neumann algebras and nuclear C

-algebras

Eric Ricard´ Jean Roydor

Abstract

We prove that von Neumann algebras and separable nuclear C^∗-algebras are stable for the Banach-Mazur cb-distance. A technical step is to show that unital almost completely isometric maps betweenC^∗-algebras are almost multiplicative and almost selfadjoint. Also as an intermediate result, we compare the Banach-Mazur cb-distance and the Kadison-Kastler distance. Finally, we show that if twoC^∗-algebras are close enough for the cb-distance, then they have at most the same length.

1 Introduction

This note concerns perturbations of operator algebras as operator spaces, more precisely pertur- bations relative to the Banach-Mazur cb-distance. In [15], G. Pisier introduced the Banach-Mazur cb-distance (or cb-distance in short) between two operator spacesX,Y:

dcb(X,Y) = inf

kTk^cbkT⁻¹k^cb ,

where the infimum runs over all possible linear completely bounded isomorphismsT:X → Y. This extends naturally the classical Banach-Mazur distance for Banach spaces when these are endowed with their minimal operator space structure (in particular, the Banach-Mazur distance and the cb-distance between two C(K)-spaces coincide). For background on completely bounded maps and operator space theory the reader is referred to [2], [8], [14] and [18].

Let us recall the generalization of Banach-Stone theorem obtained independently by D. Amir and M. Cambern (see [1], [4]): if the Banach-Mazur distance between two C(K)-spaces is strictly smaller than 2, then they are∗-isomorphic (asC^∗-algebras). Actually, this is also true for spaces of continuous functions vanishing at infinity on locally compact Hausdorff spaces. One is tempted to extend the Amir-Cambern Theorem to noncommutativeC^∗-algebras. In [12], R. Kadison de- scribed isometries betweenC^∗-algebras, in particular the isometric structure of a C^∗-algebra only determines its Jordan structure, hence to recover theC^∗-structure we need a priori assumption on the cb-distance (not only on the classical Banach-Mazur distance). Here, we prove:

Theorem A.LetAbe a separable nuclearC^∗-algebra or a von Neumann algebra, then there exists anε0>0 such that for anyC^∗-algebra B, the inequality dcb(A,B)<1 +ε0 implies that AandB are∗-isomorphic.

WhenAis a separable nuclearC^∗-algebra, one can takeε0= 3.10⁻¹⁹. WhenAis a von Neumann algebra,ε0= 4.10⁻⁶is sufficient.

Such a result can not be extended to all unital C^∗-algebras, see Corollary 3.9 below for a counter-example (derived from [3]) involving nonseparableC^∗-algebras.

The proof of Theorem A is totally different from the commutative case, the cb-distance concerns only the operator space structure, hence the basic idea is to gain the algebraic structure. It is known that unital completely isometric linear isomorphisms between operator algebras are neces- sarily multiplicative (see Theorem 4.5.13 [2]). Therefore, one wants to prove that almost completely isometric maps are almost multiplicative; in the sense that the defect of multiplicativity has small cb-norm as a bilinear map. We manage to check this by an ultraproduct argument (see Proposition 2.1) but without any explicit control on the defect of multiplicativity. When maps are betweenC^∗- algebras, we can drop the ‘unital’ hypothesis and show that the unitization of an almost completely

isometric map betweenC^∗-algebras is almost multiplicative with explicit bounds. Consequently, starting from a linear cb-isomorphism with small bound between C^∗-algebras, one can define a new multiplication on each of them close to the original ones. Then, in the spirit of [11] or [19], we use the vanishing of the second and third completely bounded Hochschild cohomology groups of an operator algebra over itself to establish a strong stability property under perturbation by close multiplications (see Proposition 3.2). It is crucial to work with the completely bounded co- homology here, because we can exploit the deep result that every completely bounded cohomology group of a von Neumann algebra over itself vanishes (see [20]), this is unknown for the bounded cohomology. This allows us to conclude for von Neumann algebras.

For separable nuclearC^∗-algebras, the strategy is different, because vanishing of completely bounded cohomology groups is not available. First, we compare the cb-distance dcb and the completely bounded Kadison-Kastler distancedKK,cb (see the definition below):

Theorem B.There exists a constantK >0, such that for anyC^∗-algebras AandB: dKK,cb(A,B)≤Kp

lndcb(A,B).

One can choose K= 3620, whendcb(A,B)<1 + 10⁻⁷.

In order to prove this theorem, we need to control explicitly the defect of selfadjointness of a unital almost completely isometric map. Then we will use stability of separable nuclearC^∗-algebras for the Kadison-Kastler distance, this is a major result in perturbation theory. Let us recall from [7] this result more precisely. As usual H denotes a Hilbert space and B(H) its bounded linear endomorphisms. LetA,B be subalgebras of B(H), the Kadison-Kastler distance betweenA and BinsideB(H) is

dKK,H(A,B) =dB(H) Ball(A), Ball(B) ,

where dB(H) denotes the Hausdorff distance and Ball(A) (respectivelyBall(B)) denotes the unit ball of A (B respectively). More generally, for two C^∗-algebras A and B, the Kadison-Kastler distance and its completely bounded version are defined as:

dKK(A,B) = inf

π,ρ,H

dKK,H π(A), ρ(B) , dKK,cb(A,B) = inf

π,ρ,H

sup

n

dKK,ℓ²n⊗H (idMn⊗π)(A),(idMn⊗ρ)(B) ,

where the infimum runs over all faithful unital ∗-representations π: A →B(H), ρ : B → B(H) on the same Hilbert space. The main result of [7] is: for any γ <42⁻¹.10⁻⁴, if Ais a separable nuclearC^∗-algebra andBis anotherC^∗-algebra, ifdKK(A,B)≤γthenAandBare∗-isomorphic.

Therefore, it is clear that the C^∗-case of Theorem A is a corollary of this last result and our Theorem B.

We already mentioned that an Amir-Cambern type theorem is false for anyC^∗-algebras, how- ever we can try to prove that someC^∗-algebraic invariants are preserved under perturbation relative to the cb-distance. The notion of length of an operator algebra has been defined by G. Pisier in [16]

in order to attack the Kadison similarity problem (he proved that aC^∗-algebra has finite length if and only if it has the Kadison similarity property). In [6] Theorem 4.4, the authors proved that having finite length is a property which is stable under perturbation for the Kadison-Kastler distance. Here, we prove that if two C^∗-algebras are close enough for the cb-distance, then they have at most the same length.

Theorem C. Let K ≥ 1 and ℓ ∈ N\{0} fixed but arbitrary constants. If A and B are unital C^∗-algebras with dcb(A,B)<1 + 10⁻⁴4^−ℓK⁻² andAhas length at most ℓ and length constant at mostK, thenB has length at mostℓ.

2 Almost completely isometric maps

This section starts with few technical lemmas relating algebraic properties to norm estimates in operator algebras. We next use them to study almost completely isometric isomorphisms. We implicitly refer to [8], [14] and [18] for basic notions around operator spaces.

It is well known that unital completely isometric isomorphisms between operator algebras are necessarily multiplicative (see Theorem 4.5.13 [2]). Hence one can hope that unital almost com- pletely isometric bijections are almost multiplicative and almost selfadjoint. This can be checked easily by an ultraproduct argument, but the important point is to control explicitly the defect of multiplicativity and the defect of selfadjointness.

WhenT :A → Bis a map between two operator algebras, the defect of multiplicativity ofT is denoted byT^∨. It consists of the bilinear mapT^∨:A²→ Bgiven byT^∨(a, b) =T(ab)−T(a)T(b).

As usual when dealing with bilinear maps (see section 1.4 in [20]), the completely bounded norm ofT^∨is the cb-norm of the induced linear mapT^∨:A ⊗^hA → Bon the Haagerup tensor product.

Given a cb mapT :S → T between two operator systems, we use the notationT^⋆for the map defined onS byT^⋆(x) =T(x^∗)^∗. We call defect of selfadjointness the linear mapT−T^⋆.

Proposition 2.1 For any η >0, there exists ρ∈]0,1[such that for any unital operator algebras A, B, for any unital cb-isomorphism T : A → B, kTk^cb ≤ 1 +ρ and kT⁻¹k^cb ≤ 1 +ρ imply kT^∨k^cb< η.

Proof: Suppose the assertion is false. Then there existsη0>0 such that for every positive integer n∈N\{0}, there is a unital cb-isomorphismTn:Aⁿ → Bⁿbetween some unital operator algebras satisfying

kTnk^cb≤1 + 1

n, kT_n⁻¹k^cb≤1 + 1

n and kT_n^∨k^cb≥η0.

LetUbe a nontrivial ultrafilter onN, let us denoteAU(resp. BU) the ultraproduct ΠnK¹⊗^minAⁿ/U (resp. ΠnK¹⊗^minBⁿ/U), hereK¹denotes the unitization of theC^∗-algebra of all compact operators onℓ2. ThenAU (resp. BU) is a unital operator algebra (see [2]). Now considerT_U:AU → BU the ultraproduct map obtained from the idK¹⊗Tn’s. Hence T_U is a unital surjective linear complete isometry between operator algebras, soT_U is multiplicative (see Theorem 4.5.13 [2]) henceT_U^∨= 0.

This contradicts the hypothesis for all n, kT_n^∨k^cb ≥η0. Indeed kT_n^∨k^cb =k(idK¹⊗Tn)^∨k, so there areun, vn in the closed unit ball ofK¹⊗^minAⁿ such that

(idK1⊗Tn)(unvn)−(idK1⊗Tn)(un)(idK1⊗Tn)(vn) ≥η0, which implies that

T_U( ˙uv)˙ −T_U( ˙u)T_U( ˙v) ≥η0

(where ˙xdenotes the equivalence class of (xn)n inAU).

A similar proof gives

Proposition 2.2 For anyη >0, there existsρ∈]0,1[such that for any operator systemsS,T, for any unital cb-isomorphismT :S → T,kTk^cb≤1 +ρandkT⁻¹k^cb≤1 +ρimplykT−T^⋆k^cb< η.

We turn to quantitative versions of the previous Propositions forC^∗-algebras.

The next Lemma is interesting because it gives an operator space characterization (it involves computations on 2×2 matrices) of invertibility inside a von Neumann algebra.

Lemma 2.3 Let Mbe a von Neumann algebra andx∈ M,kxk ≤1. Then,xis invertible if and only if there existsα >0 such that for any projection y∈ M,

x y

2

≥α+kyk² and

x y

2≥α+kyk² (C)

If this holds, the maximum of theα’s satisfying(C)equalskx⁻¹k⁻² and(C)holds for anyy∈ M. Proof: Ifx∈ Mis invertible andy∈ Mis arbitrary, by functional calculus,x^∗x≥ kx⁻¹k⁻², thus

x y

2

=kx^∗x+y^∗yk ≥

kx⁻¹k⁻²+y^∗y

=kx⁻¹k⁻²+kyk².

Thanks to a similar argument for the row estimate, we get that (C) holds withα=kx⁻¹k⁻².

Assume that (C) is satisfied. Fixλ≥0 and letpλ=χ_[0,λ](x^∗x)∈ Mbe the spectral projection of |x|² corresponding to [0, λ]. By the functional calculus 1 +λ≥x^∗x+pλ as kxk ≤ 1. Taking y=pλ in the first part of (C) gives,

1 +λ≥

x pλ

2

≥α+kpλk². Hence pλ = 0 for λ < α. Thus x^∗xis invertible and

(x^∗x)⁻¹

≤α⁻¹. Similarlyxx^∗ must have the same property andxis left and right invertible hence invertible. In the polar decomposition ofx=u|x|,umust be a unitary so that we finally getα≤ kx⁻¹k⁻²and the proof is complete.

The next Proposition generalizes the well-known fact that a complete isometry between C^∗- algebras sends unitaries to unitaries.

Proposition 2.4 Let A, B be two C^∗-algebras. Let T : A → B be a cb-isomorphism such that kTk^cbkT⁻¹k^cb<√

2. Then, for any unitaryu∈A,T(u)is invertible and kT(u)⁻¹k ≤ kT⁻¹k^cb

p2− kTk²cbkT⁻¹k²cb

.

Proof : Passing to biduals, we can assume thatA and B are von Neumann algebras asx∈ B is invertible inB if and only if it is invertible inB^∗∗. ReplacingT byT /kTk^cb, we may assume that kTk^cb= 1 andkT⁻¹k^cb<√

2.

Lety∈ Bwithkyk= 1, askT(u)k ≤1:

T(u) y

2

≥ 1

kT⁻¹k²cb

u T⁻¹(y)

2

≥1 +kT⁻¹(y)k²

kT⁻¹k²cb ≥ 2 kT⁻¹k²cb

. HenceT(u) satisfies (C) withα= _kT−1²_k2

cb−1>0. Finally applying Lemma 2.3, we obtain kT(u)⁻¹k²≤ 1

α≤ kT⁻¹k²cb

2− kT⁻¹k²cb

.

This Lemma is folklore, we give a quick proof.

Lemma 2.5 LetAbe a unitalC^∗-algebra andx∈ Ainvertible. Then there exists a unitaryu∈ A such thatkx−uk= maxn

kxk −1,1−_kx⁻¹¹ k

o.

Proof: Write the polar decomposition ofx=u|x|. Asxis invertible,|x|is strictly positive element ofA, souis a unitary ofA. Obviously,kx−uk=k|x|−1k. Seeing|x|as a strictly positive function thanks to the functional calculus, it is not difficult to conclude.

The next Lemma is the key result to compute explicitly the defect of multiplicativity. As in Lemma 2.3, operator space structure is needed.

Lemma 2.6 Let u, v be two unitaries inB(H). Let x∈B(H)andc≥1 such that

u x

−1 v

≤c√

2, then

x−uv ≤2√

c²−1.

Proof: Note first that

u^∗ 0

0 1

u x

−1 v

1 0 0 v^∗

=

1 u^∗xv^∗

−1 1

, hence without loss of generality we can assume thatu=v= 1. Takeh∈H, then

1 x

−1 1 −h h

≤c√

2

−h h

. Thereforekx(h)−hk²+ 4khk²≤4c²khk², which implieskx−1k ≤2√

c²−1.

We are now ready to prove the main result of this section.

Theorem 2.7 Let A, B be two unital C^∗-algebras. Let T : A → B be a cb-isomorphism with T(1) = 1 andkTk^cbkT⁻¹k^cb<√

2, then kT^∨k^cb≤2

s

kTk^cb+µ(T)

√2 2

−1 +µ(T)(1 +kTk^cb),

kT−T^⋆k^cb≤2 s

kTk^cb+µ(T)

√2 2

−1 + 2µ(T),

whereµ(T) = maxn

kTk^cb−1,1− s 2

kT⁻¹k²cb− kTk²cb

o.

Proof : We start with the defect of multiplicativity. From the definition of the Haagerup tensor norm and the Russo-Dye Theorem, it suffices to show that for any unitariesu, v∈M_n(A) we have

kTn(uv)−Tn(u)Tn(v)k^Mn(B)≤2 s

kTk^cb+µ(T)

√2 2

−1 +µ(T)(1 +kTk^cb), whereTn=IdMn⊗T. Without loss of generality, we can assumen= 1.

Letu, v∈ Aunitaries, as

u uv

−1 v

=√

2, we get

T(u) T(uv)

−1 T(v)

≤ kTk^cb√ 2.

From Lemma 2.5 and Proposition 2.4 we deduce that there are unitariesu^′, v^′ ∈ B withkT(u)− u^′k ≤µ(T) andkT(v)−v^′k ≤µ(T). The triangular inequality gives

u^′ T(uv)

−1 v^′

≤ kTk^cb√

2 +µ(T).

Lemma 2.6 implies that kT(uv)−u^′v^′k ≤ 2 r

kTk^cb+^µ(T√₂⁾2

−1, so that we get the estimate using the triangular inequality once more.

For the second estimate, asT_n^⋆= (T^⋆)n we may also assumen= 1. Thanks to the Russo-Dye Theorem, we just need to check that for anyu∈ Aunitary

kT(u)−T(u^∗)^∗k ≤2 s

kTk^cb+µ(T)

√2 2

−1 + 2µ(T).

Takingv=u^∗in the above arguments leads tokT(uu^∗)−u^′v^′k ≤2 r

kTk^cb+^µ(T√ ⁾ 2

2

−1. Hence ku^′−v^′∗k ≤2

r

kTk^cb+^µ(T√ ⁾ 2

2

−1 and we conclude using the triangular inequality.

3 A noncommutative Amir-Cambern Theorem

3.1 Perturbations of multiplications

Definition 3.1 Let X be an operator space. A bilinear mapm :X × X → X is called a multipli- cation onX if it is associative and extends to the Haagerup tensor product X ⊗^hX.

We denote bym_Athe original multiplication on an operator algebra A.

In the following, H_cb^k(A,A) denotes thek^th completely bounded cohomology group ofAover itself. We refer to [20] for precise definitions.

The next proposition is the operator space version of Theorem 3 in [19] or Theorem 2.1 in [11]. It gives a precise form of small perturbations of the product on an operator algebra under cohomological conditions.

As before, the quantity km−m_Ak^cbis the cb-norm of m−m_A as a linear map fromA ⊗^hA intoA.

Proposition 3.2 Let Abe an operator algebra satisfying

H_cb²(A,A) =H_cb³(A,A) = 0. (⋆)

Then there exist δ, C >0 such that for every multiplication m on Asatisfying km−m_Ak^cb≤δ, there is a completely bounded linear isomorphism Φ :A → Asuch that

kΦ−id_Ak^cb≤Ckm−m_Ak^cb and Φ(m(x, y)) = Φ(x)Φ(y).

IfAis a von Neumann algebra, then(⋆)is automatically satisfied with valuesδ= 1/11andC= 10.

Moreover ifm satisfiesm(x^∗, y^∗) = m(y, x)^∗ for allx, y∈ A, thenΦ(x^∗) = Φ(x)^∗ for allx∈ A. Proof: We only give a sketch as it only consists in adapting arguments of [19] (see also [20] chapter 7) or Theorem 2.1 of [11] to the operator space category.

In the bounded situation, one has to apply an implicit function theorem (Theorem 1 in [19]) to the right spaces of multilinear maps (Theorem 3 in [19]). This is done in details in Theorem 7.4.1 in [20] from 7.3.1. With the notation there (taking M = A) one simply need to replace L^k(M,M) by their cb-version L^k_cb(A,A) which are obviously Banach spaces. The statement about

∗is justified right after Theorem 7.4.1 in [20].

IfAis a von Neumann algebra, all completely bounded cohomology groups ofAover itself vanish (see [20] chapter 4.3) and we can chooseK=L= 1 in the proof of Theorem 2.1 of [11], which gives δ= 11⁻¹(see the discussion after the proof of Theorem 2.1 [11]) and a computable value ofC. Now following notation of the proof of Theorem 2.1 [11], the rational functionpsatisfiesp(x)≤9.75x², for x small enough. Hence the sequence (εi) defined by εi+1 = p(εi) (and ε0 = km−m_Ak^cb) verifiesεi≤9.75²ⁱ⁻¹ε²₀ⁱ. AsK=L= 1, we havekSik^cb≤εi−1+ 2ε²_i−1. Then, using the previous estimates of theεi’s we get kWn−Ik^cb ≤exp(Pn

i=1kSik^cb)≤10ε0.(With notation of Theorem 2.1 [11], the desired Φ is obtained as the limit of (Wn)n.)

3.2 Proofs of Theorems A and B

The proof of Theorem A is a variant of the proof of Theorem B. For clarity we postpone the quantitative estimate to the next Remark.

Proof of Theorem B: AsdKK is bounded, the statement is only interesting whendcb(A,B) is close to 1. The proof uses ideas from Corollary 7.4.2 in [20].

LetL:A → Bbe a cb-isomorphism withkLk^cb≤1 andkL⁻¹k^cb≤dcb(A,B)(1 +ǫ).

Consider the bidual extension still denoted byL:A^∗∗ → B^∗∗, it remains a cb-isomorphism and satisfies the same norm estimates.

We suppose Aunital, we will treat the non-unital case afterwards. The first step is to unitize L. By Lemma 2.4,L(1) is invertible in B. LetS =L(1)⁻¹L, thenS is unital and

kSk^cb≤ kL⁻¹k^cb p2− kL⁻¹k²cb

andkS⁻¹k^cb≤ kL⁻¹k^cb. Note also thatS(A) =B.

The second step is to make our cb-isomorphism selfadjoint. By Theorem 2.7, we have kS− S^⋆k^cb ≤2f1 kL⁻¹k^cb

for some continuous function with f1(1) = 0. Let T = ¹₂ S+S^⋆ . Then T :A^∗∗ → B^∗∗ is unital∗-preserving, kT −Sk^cb ≤f1 kL⁻¹k^cb

and T(A)⊂ B. So ifkL⁻¹k^cb is close enough to 1, T is also a cb-isomorphism such thatT(A) =B with norm estimateskTk^cb≤ f2 kL⁻¹k^cb

andkT⁻¹k^cb≤f2 kL⁻¹k^cb

, for some continuous function at 1 with f2(1) = 1.

Define onA^∗∗ a new multiplication by, forx, y∈ A^∗∗

m(x, y) =T⁻¹ T(x)T(y) .

The multiplication m is associative and ∗-preserving. It is obviously completely bounded and clearly

km−m_A^∗∗k^cb≤ kT⁻¹k^cbkT^∨k^cb.

Thus, the estimate in Theorem 2.7 gives thatkm−m_A^∗∗k^cb≤f3 kL⁻¹k^cb

for some continuous functionf3withf3(1) = 0. IfkL⁻¹k^cbis close enough to 1, we get from Proposition 3.2, that there is a completely bounded ∗-preserving linear isomorphism Φ : A^∗∗ → A^∗∗ with kΦ−id_A^∗∗k^cb ≤ f4 kL⁻¹k^cb

and forx, y∈ A^∗∗

Φ⁻¹ Φ(x)Φ(y)

= m(x, y) =T⁻¹(T(x)T(y)).

Note that necessarily Φ(1) = 1, and Φ is∗-preserving.

Let π =TΦ⁻¹ : A^∗∗ → B^∗∗, it is a ∗-preserving cb-isomorphism. Moreover, forx, y ∈ A^∗∗, π(xy) =π(x)π(y), henceπis actually a ∗-isomorphism. Now we check that the C^∗-algebrasπ(A) andBare close for the Kadison-Kastler distance insideB^∗∗. We have, fora∈Ball(A):

kπ(a)−T(a)k ≤f4 kL⁻¹k^cb

, (1)

and asT(A) =B, forb∈Ball(B), we have kb−π T⁻¹(b)

k ≤ kT⁻¹k^cbf4 kL⁻¹k^cb

=f5 kL⁻¹k^cb . From which one easily deduces dKK(A,B)≤f5 kL⁻¹k^cb

, for some continuous function f5 with f5(1) = 0.

Now if A is non-unital, in the preceding proof,L(1) is now invertible in B^∗∗ (here 1 denotes the unit ofA^∗∗), so S(A) = Bis not valid anymore. But the inequality (1) above still holds and we deduce that fora∈Ball(A)

kπ(a)−S(a)k ≤(f4+f1) kL⁻¹k^cb .

Now from Lemma 2.5, there is a unitaryuinB^∗∗such thatku−L(1)⁻¹k ≤f6 kL⁻¹k^cb

for some continuous functionf6withf6(1) = 0. Therefore

kπ(a)−uL(a)k ≤(f4+f1+f6) kL⁻¹k^cb . Taking the adjoints we obtain

kπ(a)−L(a^∗)^∗u^∗k ≤(f4+f1+f6) kL⁻¹k^cb . Writea=xy, for somexandyin the Ball(A), then

kπ(a)−uL(x)L(y^∗)^∗u^∗k ≤2(f4+f1+f6) kL⁻¹k^cb

. (2)

As L(x)L(y^∗)^∗ belongs to Ball(B), we conclude that the C^∗-algebraπ(A) is nearly included in theC^∗-algebrauBu^∗. Let us prove the converse near inclusion. Letb∈Ball(B), we can factorize b=L(x)L(y^∗)^∗ withx, y ∈ Asuch that kxk ≤ kL⁻¹k andkyk ≤ kL⁻¹k. From inequality (2), we get

kπ(xy)−uL(x)L(y^∗)^∗u^∗k ≤2kL⁻¹k²(f4+f1+f6) kL⁻¹k^cb . Finally,dKK(A,B)≤2kL⁻¹k²(f4+f1+f6) kL⁻¹k^cb

.

Proof of Theorem A: WhenAis a separable nuclearC^∗-algebra, this follows directly from Theorem 4.3 in [7] and Theorem B.

WhenA is a von Neumann algebra, one does not need to go to the bidualA^∗∗ in the preceding proof (to apply Proposition 3.2), so that we directly conclude thatπ:A → B is a∗-isomorphism.

But we should mention another way (which improves theoretically our bound in the von Neumann algebras case): the second step in the proof is not necessary, just define directly a new multiplication using the cb-isomorphismS(instead ofT). Thenπis just an algebra isomorphism (not necessarily selfadjoint), but it is enough to conclude thanks to Theorem 3 in [9].